All Questions
230 questions
6
votes
2
answers
507
views
Concerning the dimension of a complex variety modulo a prime
Let V be a complex affine variety given as the vanishing set of a set of polynomials with integral coefficients. I have 3 questions.
1)
Under what assumption will the dimension of V over C remain ...
- 63
6
votes
1
answer
825
views
More on universal homeomorphisms
I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers).
Is a universal ...
- 16.9k
6
votes
1
answer
1k
views
Generic Smoothness Type of Results in Positive Characteristic
Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth.
We know that ...
- 313
6
votes
1
answer
507
views
Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian
It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $\operatorname{char}(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-...
- 679
6
votes
1
answer
693
views
Nagata's conjecture in positive characteristic
For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ irreducible reduced curve passes then $d^...
- 5,063
6
votes
1
answer
393
views
finite quotients of fundamental groups in positive characteristic
For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental ...
- 4,265
6
votes
0
answers
173
views
Orlik-Solomon algebra and hyperplane complements in positive characteristic
Let $k$ be an algebraically closed field of characteristic $p\geq 0$, $\underline H:=\{H_1,\dots, H_m\}$ a set of hyperplanes in $\mathbb A_k^n$ and $X:=\mathbb A^n-(\bigcup H_i)$.
Given a ring $R$ ...
- 728
6
votes
0
answers
343
views
Are all stabilizer groups of the co-adjoint action smooth?
Let $k$ be a (non-archimedean) local field of positive characteristic $p$ and $\mathfrak{n}$ be any finite-dimensional nilpotent Lie algebra over $k$ with nilpotence length $l<p$. It is well-known ...
- 1,702
6
votes
0
answers
467
views
Torsionfree crystalline cohomology implies torsionfree etale cohomology?
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$. Let $W=W(k)$ be the ring of Witt vectors of $k$.
Assume that the crystalline cohomology $H^2_{...
- 91
6
votes
0
answers
436
views
Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?
As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on '...
- 16.9k
5
votes
2
answers
1k
views
Weil Conjectures for Grassmannians
To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?
- 1,462
5
votes
3
answers
739
views
Smoothness of hyperplane sections
Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}_p)$ for ...
- 13.1k
5
votes
1
answer
1k
views
Minimal Model Program for surfaces over algebraically closed fields of characteristic p
Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...
- 2,215
5
votes
1
answer
514
views
Lifting torsors in characteristic $p$ to characteristic zero
Let $R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and $k$ is of characteristic $p>0$. Let $G$ be a smooth finite type affine group scheme with ...
- 151
5
votes
1
answer
230
views
Do regular (but non-smooth) conics over a discretely valued field of characteristic $2$ admit a regular model over the valuation ring?
Let $K$ be a non-perfect field of characteristic $2$. Let $T \subseteq K$ be a discrete valuation ring.
Assume there exist $a,b \in K^{\times}$ such that the projective conic $C$ defined by $$aX^2 + ...
- 103
5
votes
1
answer
273
views
Singularities of curves that are moving
Let $k$ be an algebraically closed field, let $d\ge 2$ be an integer and let $f,g\in k[x,y,z]$ be two homogeneous polynomials of degree $d$ without common factor.
We want to know what are the ...
- 7,722
5
votes
1
answer
710
views
Log resolutions on surfaces and 3-folds in characteristic p
If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the ...
- 2,215
5
votes
2
answers
556
views
Existence of certain identities involving characteristic 2 "thetas"
Let l=2m+1 be prime. In my previous MO question, "What are the polynomial relations between these characteristic 2 thetas?", I defined a subring of Z/2[[x]] as follows:
The subring, S, is generated ...
- 5,422
5
votes
1
answer
419
views
Lifting $\mathfrak{sl}_2$-triples
Let
$k$ be an algebraically closed field,
$G$ a (smooth, connected) reductive algebraic group over $k$,
$H$ a (smooth, connected) reductive group of semisimple rank 1, and
$T$ a maximal torus in $H$.
...
- 12.9k
5
votes
2
answers
586
views
Quotient of a reductive group by a non-smooth subgroup
This is a continuation of my question Quotient of a reductive group by a non-smooth central finite subgroup.
Let $G$ be a smooth, connected, reductive $k$-group over a field $k$ of characteristic $p&...
- 14.2k
5
votes
1
answer
461
views
Given a branched cover with branch cycle description $(g_1,...,g_r)$, does $g_i$ generate some decomposition group?
Classically:
Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so $\alpha_1...\...
- 5,929
5
votes
1
answer
1k
views
Excellent schemes
I noticed that many results in positive characteristic assumes that the object of the theorem is excellent. I have looked up the definition of excellent and have tried to get a feeling for it, but all ...
- 53
5
votes
0
answers
139
views
Liouville property in the Bost theorem on foliations
Let $X$ be a smooth algebraic variety over a number field $K$, and let $\mathcal{F}$ be an involutive coherent subsheaf of the tangent bundle. After a pullback to $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
- 485
5
votes
0
answers
197
views
Semisimplicity of the p-adic étale Tate module over $F_p(t)$
Let $k$ be a finitely generated field of positive characteristic p.
Let $A$ be an abelian variety over $k$ and write $T_p(A)$ for the $p$-adic étale Tate module of $A$. Is it known if the natural ...
- 728
5
votes
0
answers
243
views
Map associated to linear system onto curve is morphism
In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...
- 61
5
votes
0
answers
568
views
Eisenbud-Goto conjecture in Positive Characteristic
Eisenbud-Goto conjecture predicted that the Castelnuovo-Mumford regularity
${\rm reg}(X)$ of a non-degenerate projective variety $X\subset \mathbb{P}^N$
is bounded by the $\deg(X)-{\rm codim}(X,\...
- 1,595
5
votes
0
answers
287
views
Nori fundamental group and etale fundamental group in positive characteristic
Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?
- 163
4
votes
1
answer
428
views
p-torsion in the Picard group of a regular projective curve
Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$.
If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of ...
- 2,051
4
votes
1
answer
272
views
How do non-trivial global differentials give non-trivial cohomology classes in positive characteristic
Let $k$ be an algebraically closed field and let $X$ be an $n$-dimensional smooth projective variety over $k$.
If $k= \mathbb C$, there is a natural injective morphism of vector spaces
$$H^0(X,\...
- 49
4
votes
2
answers
918
views
Katz's proof of Cartier's (descent) theorem
I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of ...
- 121
4
votes
2
answers
339
views
Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?
Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?
- 16.9k
4
votes
1
answer
502
views
Motivic integration in positive characteristic: how much is known?
It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...
- 16.9k
4
votes
1
answer
412
views
F_q-structures on schemes
Let $k|\mathbb{F}_q$ be a field extension. An $\mathbb{F}_q$-structure on a $k$-algebra $A$ is an $\mathbb{F}_q$-subalgebra $A _0$ of $A$ such that $A _0 \otimes _{\mathbb{F}_q} k \cong A$ via the ...
- 5,243
4
votes
2
answers
340
views
On families of supersingular abelian surfaces over the projective line
Let $k=\mathbb{F}_q$. I recently learned that there are non-isotrivial families $f:X\to \mathbb{P}^1_k$ of supersingular abelian surfaces. In particular, the Kodaira-Spencer map of this family is non-...
- 43
4
votes
1
answer
358
views
Examples of perfect pseudo algebraically closed fields in positive characteristic
Is there any known example of a perfect pseudo algebraically closed field of positive characteristic containing $\overline{\mathbb{F}_p}$ but is not algebraically closed?
- 2,323
4
votes
1
answer
398
views
Properties of divisors when moving from char 0 to char p.
Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
- 141
4
votes
1
answer
369
views
Gauss mapping in finite characteristic
Suppose that $X\subset\mathbb P^n$ is a $d$-dimentional smooth projective variety (not a linear subspace) over an algebraically closed field. If $\gamma\colon X\to\mathrm{Gr}(d,\mathbb P^n)$ is Gauss ...
- 1,939
4
votes
2
answers
694
views
Ample line bundle and Frobenius morphism on smooth toric variety
Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $X$ a smooth toric projective variety of $\dim X=n$, $F_X:X\rightarrow X$ the absolute Frobenius morphism of $X$. Then for any $\...
- 85
4
votes
1
answer
272
views
Finiteness of cohomology with finite coefficients
Let $G$ be a finite abelian group and let $S$ be a variety over $\mathbb{F}_p$. It is natural (I think) to expect that the cohomology group $H^i(S,G)$ is finite. But with respect to which cohomology?
...
- 41
4
votes
1
answer
796
views
Deformation space of non-ordinary abelian varieties
It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian ...
- 395
4
votes
2
answers
460
views
Are Chow groups invariant under universal homeomorphisms?
Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...
- 16.9k
4
votes
1
answer
622
views
finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme
Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.
Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{...
user19475
4
votes
0
answers
108
views
Shafarevich conjecture for Abelian varieties over global function fields
Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
- 679
4
votes
0
answers
103
views
Boundary divisor of an equivariant compactification of a non-trivial twist of $\mathbb{G}_a^n$
Let $K$ be a non-perfect field of positive characteristic. Then there exist non-trivial forms of $\mathbb{G}_a^n$ which split over finite purely inseparable extensions of $K$, i.e., algebraic groups $...
4
votes
0
answers
296
views
de Rham Witt complex vs. de Rham complex of the Witt ring
I am reading the paper "Revisiting the de Rham-Witt complex" by Bhatt-Lurie-Mathew and I am a bit confused about the difference between $W\Omega_R^*$ and $\hat{\Omega}^*_{W(R)}$.
Let $\...
- 218
4
votes
0
answers
204
views
Explicit description of wonderful compactification for PGL_3
Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry ...
- 163
4
votes
0
answers
215
views
Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence
My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...
- 485
4
votes
0
answers
189
views
If the Frobenius endomorphism of a characteristic $p$ ring is epimorphic, is it surjective?
MO question 19282 is about integral epimorphisms of commutative rings, and a counterexample is given to surjectivity. What about the case of the Frobenius endomorphism of a commutative, characteristic ...
- 4,492